CLAED7 used by sstedc. Computes the updated eigensystem of a diagonal matrix after modification by a rank-one symmetric matrix. Used when the original matrix is dense.

Purpose:

CLAED7 computes the updated eigensystem of a diagonal
matrix after modification by a rank-one symmetric matrix. This
routine is used only for the eigenproblem which requires all
eigenvalues and optionally eigenvectors of a dense or banded
Hermitian matrix that has been reduced to tridiagonal form.
T = Q(in) ( D(in) + RHO * Z*Z**H ) Q**H(in) = Q(out) * D(out) * Q**H(out)
where Z = Q**Hu, u is a vector of length N with ones in the
CUTPNT and CUTPNT + 1 th elements and zeros elsewhere.
The eigenvectors of the original matrix are stored in Q, and the
eigenvalues are in D. The algorithm consists of three stages:
The first stage consists of deflating the size of the problem
when there are multiple eigenvalues or if there is a zero in
the Z vector. For each such occurrence the dimension of the
secular equation problem is reduced by one. This stage is
performed by the routine SLAED2.
The second stage consists of calculating the updated
eigenvalues. This is done by finding the roots of the secular
equation via the routine SLAED4 (as called by SLAED3).
This routine also calculates the eigenvectors of the current
problem.
The final stage consists of computing the updated eigenvectors
directly using the updated eigenvalues. The eigenvectors for
the current problem are multiplied with the eigenvectors from
the overall problem.

Parameters:

N

N is INTEGER
The dimension of the symmetric tridiagonal matrix. N >= 0.

CUTPNT

CUTPNT is INTEGER
Contains the location of the last eigenvalue in the leading
sub-matrix. min(1,N) <= CUTPNT <= N.

QSIZ

QSIZ is INTEGER
The dimension of the unitary matrix used to reduce
the full matrix to tridiagonal form. QSIZ >= N.

TLVLS

TLVLS is INTEGER
The total number of merging levels in the overall divide and
conquer tree.

CURLVL

CURLVL is INTEGER
The current level in the overall merge routine,
0 <= curlvl <= tlvls.

CURPBM

CURPBM is INTEGER
The current problem in the current level in the overall
merge routine (counting from upper left to lower right).

D

D is REAL array, dimension (N)
On entry, the eigenvalues of the rank-1-perturbed matrix.
On exit, the eigenvalues of the repaired matrix.

Q

Q is COMPLEX array, dimension (LDQ,N)
On entry, the eigenvectors of the rank-1-perturbed matrix.
On exit, the eigenvectors of the repaired tridiagonal matrix.

LDQ

LDQ is INTEGER
The leading dimension of the array Q. LDQ >= max(1,N).

RHO

RHO is REAL
Contains the subdiagonal element used to create the rank-1
modification.

INDXQ

INDXQ is INTEGER array, dimension (N)
This contains the permutation which will reintegrate the
subproblem just solved back into sorted order,
ie. D( INDXQ( I = 1, N ) ) will be in ascending order.

QPTR is INTEGER array, dimension (N+2)
List of indices pointing to beginning of submatrices stored
in QSTORE. The submatrices are numbered starting at the
bottom left of the divide and conquer tree, from left to
right and bottom to top.

PRMPTR

PRMPTR is INTEGER array, dimension (N lg N)
Contains a list of pointers which indicate where in PERM a
level's permutation is stored. PRMPTR(i+1) - PRMPTR(i)
indicates the size of the permutation and also the size of
the full, non-deflated problem.

PERM

PERM is INTEGER array, dimension (N lg N)
Contains the permutations (from deflation and sorting) to be
applied to each eigenblock.

GIVPTR

GIVPTR is INTEGER array, dimension (N lg N)
Contains a list of pointers which indicate where in GIVCOL a
level's Givens rotations are stored. GIVPTR(i+1) - GIVPTR(i)
indicates the number of Givens rotations.

GIVCOL

GIVCOL is INTEGER array, dimension (2, N lg N)
Each pair of numbers indicates a pair of columns to take place
in a Givens rotation.

GIVNUM

GIVNUM is REAL array, dimension (2, N lg N)
Each number indicates the S value to be used in the
corresponding Givens rotation.